Final answer:
To express the polynomial as a product of linear factors, divide it by x + 3 to find the quadratic part, then use the quadratic formula to find the remaining zeros, and represent h(x) as the product of all linear factors corresponding to its zeros.
Step-by-step explanation:
The student has given a polynomial h(x) = x³ - x² + 17x + 87 and stated that -3 is a zero of h(x). To express h(x) as a product of linear factors, the polynomial division or synthetic division can be used to divide the polynomial by x + 3 (since -3 is a zero). After finding the quotient, which will be a quadratic polynomial, we can use the quadratic formula to find the remaining zeros of the polynomial.
Once we have all the zeros, h(x) can be expressed as the product of linear factors that correspond to these zeros. If 'r' is a zero of the polynomial, then x - r will be one of its linear factors. Employing all the zeros found, we would multiply them in the form (x - zero1)(x - zero2)(x - zero3) to get h(x) in fully factored form.